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<p>The general form of ODE is</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq1_5.html ./knowl/eq1_5.html ./knowl/eq1_5.html">
\begin{equation}
F(x, y, y^{\prime}, y^{\prime \prime}, \cdots, y^{(n)})=0.\tag{1.2.1}
\end{equation}
</div>
<p class="continuation">If <span class="process-math">\(F(\cdots)\)</span> is a linear function of <span class="process-math">\(y^{\prime}, \cdots, y^{(n)}\text{,}\)</span> i.e., (<a href="" class="xref" data-knowl="./knowl/eq1_5.html" title="Equation 1.2.1">(1.2.1)</a>) takes the form</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq1_5.html ./knowl/eq1_5.html ./knowl/eq1_5.html">
\begin{equation}
a_0(x) y^{(n)}+a_1(x) y^{(n-1)}+\cdots+a_{n-1}(x) y^{\prime}+a_n(x) y=q(x),\tag{1.2.2}
\end{equation}
</div>
<p class="continuation">then, we say that (<a href="" class="xref" data-knowl="./knowl/eq1_5.html" title="Equation 1.2.1">(1.2.1)</a>) is a linear ODE. Otherwise, (<a href="" class="xref" data-knowl="./knowl/eq1_5.html" title="Equation 1.2.1">(1.2.1)</a>) is said to be a nonlinear ODE. For example, <span class="process-math">\(\frac{\textrm{d}y}{\textrm{d}x}=x\)</span> is linear while <span class="process-math">\(\frac{\textrm{d}^2\theta}{\textrm{d} t^2}+\sin \theta=0\)</span> is nonlinear.</p>
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